TIFR GS 2017 Question Paper Mathematics

MTH
GS-2017 (Mathematics)
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
Written Test in MATHEMATICS ‐ December 11, 2016
For the Ph.D. Programs at TIFR, Mumbai and CAM & ICTS, Bangalore and for the
Int. Ph.D. Programs at TIFR, Mumbai and CAM, Bangalore.

Duration : Three hours (3 hours)
Name : _______________________________________________ Ref. Code : ____________

Please read all instructions carefully before you attempt the questions.

1. Please fill in details about name, reference code etc. on the answer sheet for Part I as well on
the answer booklet of Part II . The Answer Sheet for Part I is machine-readable. Use only
Black/Blue ball point pen to fill in the answer sheet.

2. PART I - There are thirty (30) True/False type questions in Part I of the question paper. Allotted
time for Part I is 90 minutes. The answer sheet for Part I will be collected at the end of 90
minutes. Part I questions carry +2 for a correct answer, -1 (negative marks) for a wrong answer
and 0 for not answering.

Indicate your answer ON THE ANSWER SHEET by blackening the appropriate circle for each
question. Do not mark more than one circle for any question : this will be treated as a wrong
answer.

We advise you to first mark the correct answers on the QUESTION PAPER and then to
TRANSFER these to the ANSWER SHEET only when you are sure of your choice.

3. PART II – 10 problems to be solved. The solutions should be written in the Answer Booklet for
Part II that is provided. Extra blank sheets will be provided if needed. All Part II questions carry
equal marks, and there are no negative marks. Partial credit will be given for partial solutions.

Candidate can begin answering questions on Part II anytime. The answer booklet for Part II will
be collected at the end of the exam.

4. Selection Procedure : The answers for Part I will be machine-graded. Part I will score will be
used to decide a cut-off. Answer papers for Part II will be graded only for those candidates
whose score is above the cut-off. List of candidates to be called for interview for the final
selection for admission in the various programs will be decided based on the combined
performance in Part I and II, weighted appropriately for each program.

5. Rough work may be done on blank pages of the question paper. If needed, you may ask for
extra rough sheets from an invigilator.

6. Use of calculators, mobile phones, laptops, tablets (or other electronic devices, including
those connecting to the internet) is NOT permitted.

7. Do NOT ask for clarifications from the invigilators regarding the questions. They have been
instructed not to respond to any such inquiries from candidates. In case a correction/clarification
is deemed necessary, the invigilators will announce it publicly.

8. Notation and Conventions used in this test are given on page 2 of the question paper.

1

Mathematics Question Paper, GS2017
Parts I and II

Notation and Conventions:

• N denotes the set of natural numbers {0, 1, 2, 3, · · · }, Z the set of inte-
gers, Q the set of rationals, R the set of real numbers and C the set of
complex numbers. These sets are assumed to carry the usual algebraic
and metric structures.

• Rn denotes the Euclidean space of dimension n. Subsets of Rn are
assumed to carry the induced topology and metric. For a vector v =
(v1 , v2 , · · · , vn ) ∈ Rn , the norm ||v|| is defined by ||v||2 = v12 + · · · + vn2 .

• Mn (R) denotes the real vector space of n × n real matrices with the
Euclidean metric.

• All logarithms are natural logarithms.

Part I
Answer whether the following statements are True or False. Mark your
answer on the machince checkable answer sheet that is provided.
Note: +2 marks for a correct answer, −1 mark (negative marks) for a
wrong answer, 0 marks for not answering.

1. Let f : [0, 1] → R be a continuous function such that f (x) ≥ x3 for all
1 True
x ∈ [0, 1] with 0 f (x)dx = 14 . Then f (x) = x3 for all x ∈ R.

2. Suppose a, b, c are positive real numbers such that
 
1 1 1
(1 + a + b + c) 1 + + + = 16.
a b c

Then a + b + c = 3.

3. There exists a function f : R → R satisfying, False
3
f (−1) = −1, f (1) = 1 and |f (x)−f (y)| ≤ |x−y| , 2 for all x, y ∈ R.



4. Over the real line, False
 √ √ 
lim log 1+ 4+x− 1 + x = log (2).
x→∞

5. Suppose f is a continuously differentiable function on R such that False
f (x) → 1 and f  (x) → b as x → ∞. Then b = 1.

6. If f : R → R is differentiable and bijective, then f −1 is also differen- False
tiable.

7. Let H1 , H2 , H3 , H4 be four hyperplanes in R3 . The maximum pos-
sible number of connected components of R3 −(H1 ∪H2 ∪H3 ∪H4 ) is 14.

8. Let n ≥ 2 be a natural number. Let S be the set of all n × n real ma-
trices whose entries are only 0, 1 or 2. Then the average determinant
of a matrix in S is greater than or equal to 1.

9. For any metric space (X, d) with X finite, there exists an isometric
embedding f : X → R4 .

10. There exists a non-negative continuous function f : [0, 1] → R such False
1
that 0 f n dx → 2 as n → ∞.

11. There exists a subset A of N with exactly five elements such that the
sum of any three elements of A is a prime number.

12. There exists a finite abelian group G containing exactly 60 elements of
order 2.

13. Let α, β be complex numbers with non-positive real parts. Then
|eα − eβ | ≤ |α − β|.

14. Every 2 × 2-matrix over C is a square of some matrix. False



15. Under the projection map R2 → R sending (x, y) to x, the image of False
any closed set is closed.

16. The number of ways a 2 × 8 rectangle can be tiled with rectangular
tiles of size 2 × 1 is 34.

17. Over the real line,  x
x + log 9 True
lim = 81.
x→∞ x − log 9

18. Let f : [0, ∞) → R be a continuous function with limx→∞ f (x) = 0. False
Then f has a maximum value in [0, ∞).

19. Given a continuous function f : Q → Q, there exists a continuous func-
tion g : R → R such that the restriction of g to Q is f .

20. For all positive integers m and n, if A is an m × n real matrix, and B False
is an n × m real matrix such that AB = I, then BA = I.

21. There is a continuous onto function f : S 2 → S 1 from the unit sphere
in R3 to the unit sphere in R2 , where S n = {v ∈ Rn+1 | ||v|| = 1}
denotes the unit sphere in Rn+1 .

22. Let P be a monic, non-zero, polynomial of even degree, and K > 0.
Then the function P (x) − Kex has a real zero.

23. A p-Sylow subgroup of the underlying additive group of a finite com-
mutative ring R is an ideal in R.

24. Suppose A is an n × n-real matrix, all whose eigenvalues have absolute
value less than 1. Then for any v ∈ Rn , ||Av|| ≤ ||v||. False

25. For any x ∈ R, the sequence {an }, where a1 = x and an+1 = cos(an )
for all n, is convergent.

26. Suppose A1 , · · · , Am are distinct n×n real matrices such that Ai Aj = 0 False
for all i = j. Then m ≤ n.



27. In the symmetric group Sn any two elements of the same order are False
conjugate.

28. If a particle moving on the Euclidean line traverses distance 1 in time
1 starting and ending at rest, then at some time t ∈ [0, 1], the absolute
value of its acceleration should be at least 4.

29. Let y(t) be a real valued function defined on the real line such that
y  = y(1 − y), with y(0) ∈ [0, 1]. Then limt→∞ y(t) = 1.

30. The matrices    
x 0 x 1
0 y
and
0 y
, x = y, True

for any x, y ∈ R are conjugate in M2 (R).

Part II
Write your solutions in the answer booklet provided. All questions carry
equal marks. There are no negative marks, and partial credit will be given
for partial solutions.

1. Show that the subset GLn (R) of Mn (R) consisting of all invertible ma-
trices is dense in Mn (R).

2. Let f be a continuous function on R satisfying the relation

f (f (f (x))) = x for all x ∈ R.

Prove or disprove that f is the identity function.

3. Prove or disprove: the group of positive rationals under multiplication
is isomorphic to its subgroup consisting of rationals which can be ex-
pressed as p/q, where both p and q are odd positive integers.

4. Show that the only elements in Mn (R) commuting with every idempo-
tent matrix are the scalar matrices. (A matrix P in Mn (R) is said to
be idempotent if P 2 = P .)



5. Prove or disprove the following: let f : X → X be a continuous
function from a complete metric space (X, d) into itself such that
d(f (x), f (y)) < d(x, y) whenever x = y. Then f has a fixed point.

6. How many isomorphism classes of associative rings (with identity) are
there with 35 elements? Prove your answer.

7. Prove or disprove: If G is a finite group and g, h ∈ G, then g, h have
the same order if and only if there exists a group H containing G such
that g and h are conjugate in H.

8. Prove or disprove: there exists A ⊂ N with exactly five elements, such
that sum of any three elements of A is a prime number.

9. Show that there does not exist any continuous function f : R → R that
takes every value exactly twice.

10. For which positive integers n does there exist a R-linear ring homomor-
phism f : C → Mn (R)? Justify your answer.